False membership rate control in mixture models

TL;DR

This paper develops a method for controlling the false membership rate in mixture models, ensuring misclassification stays below a pre-set level, with theoretical guarantees and improved bootstrap-based procedures.

Contribution

It introduces a novel plug-in method with theoretical analysis for FMR control in unsupervised mixture models, including bootstrap enhancements.

Findings

The proposed method guarantees FMR does not exceed the nominal level.

Bootstrap procedures improve the accuracy and robustness of FMR control.

Theoretical bounds quantify the deviation of FMR from the target level.

Abstract

The clustering task consists in partitioning elements of a sample into homogeneous groups. Most datasets contain individuals that are ambiguous and intrinsically difficult to attribute to one or another cluster. However, in practical applications, misclassifying individuals is potentially disastrous and should be avoided. To keep the misclassification rate small, one can decide to classify only a part of the sample. In the supervised setting, this approach is well known and referred to as classification with an abstention option. In this paper the approach is revisited in an unsupervised mixture model framework and the purpose is to develop a method that comes with the guarantee that the false membership rate (FMR) does not exceed a pre-defined nominal level $\alpha$. A plug-in procedure is proposed, for which a theoretical analysis is provided, by quantifying the FMR deviation with respect to the target level $\alpha$ with explicit remainder terms. Bootstrap versions of the procedure are shown to improve the performance in numerical experiments.